Educational Statistics Activities of Statisticians Major activities in

Educational Statistics: Activities of Statisticians • Major activities in statistics involve: – design of experiments and surveys to test hypotheses – exploration and visualization of sample data – summary description of sample data – modeling of uncertainty (e. g. flipping a coin) – forecasting based on suitable models – hypothesis testing and statistical inference

Educational Statistics: Very Brief History • Chevalier de Mėre’s gambling problems and Blaise Pascal’s solutions (mid-1600 s). • Abraham de Moivre’s publication (in English) of his Doctrine of Chance in mid 1700 s. • William Sealy Gossett’s development of the formula, in the early 1900 s, for the standard error of the mean. • Development of the t-test, analysis of variance, and non-parametric statistics in the first quarter of the 1900 s.

Educational Statistics: Statistical Terms and Vocabulary • Statistics: a set of methods, procedures and rules for organizing, summarizing, and interpreting information. – This is a general definition. – Later, a distinction between statistics and parameters will be made. – Here, it would be better to speak of statistical methods.

Educational Statistics: Statistical Terms and Vocabulary • Use of symbols in statistics – Statisticians (and statistical books) use symbols as shorthands for complex concepts and constructs. – Symbols are typically either Arabic or Greek letters. – For example: • µ (the Greek letter, mu) typically represents the mean (arithmetic average) of a set of values. • σ (the lower-case Greek letter, sigma) typically represents the standard deviation of a set of values. • For a list of Greek symbols and their meanings in statistics, see Symbols and Notation under General links to a variety of sources in the Table of Contents.

Educational Statistics: Statistical Terms and Vocabulary Three Types of statistical methods: • Descriptive statistics: methods used to summarize, organize, and simplify data. • Exploratory statistics: methods for carefully examining data prior to using more complicated statistical procedures. • Inferential statistics: methods that allow us to make generalizations about populations based on data obtained from samples.

Educational Statistics: Statistical Terms and Vocabulary Population vs Sample • Population: all members of a particular group (e. g. , all Appstate freshman, all males over the age of 21, all of the schools in NC). • Sample: a subgroup of a population that is usually assumed to be representative of the population (e. g. , 10 Appstate freshman selected at random).

Educational Statistics: Statistical Terms and Vocabulary Parameters and Statistics • Parameter: the value of a variable in a population. • Statistic: the value of a variable in a sample. • Statistics are often used to estimate or draw inferences about parameters.

Educational Statistics: Statistical Terms and Vocabulary Variable: any characteristic that can vary across individuals, groups, or objects. For example: • Weight • Occupation • Grade-point average • Level of test anxiety Later we will look at various types of variables.

Educational Statistics: Statistical Terms and Vocabulary Values: the numerical value of a particular realization of a variable. For instance if the variable is weight and Mortimer weighs 147 lbs. Then the value of the variable for Mortimer is 147. Make sure you can distinguish between variables and values

Educational Statistics: Statistical Terms and Vocabulary Sampling error: the difference between a sample statistic and its corresponding population parameter. • The values of sample statistics vary from sample to sample, even when all samples are drawn from the same population.

Educational Statistics: Statistical Terms and Vocabulary Distributions • Organized arrangements of sets of data by order of magnitude. – Sequential listings of data points from lowest to highest. – Frequency distributions. • A sequential listing of data points combined with the number of times (or frequency with which) each point occurs.

Educational Statistics: Statistical Terms and Vocabulary Two important types of distributions: – Empirical distributions, e. g. , Distributions of test scores. Distributions of weight, height, etc. – Theoretical distributions, e. g. , Distribution of GRE scores for ALL ASU graduate students. Statistical distributions.

Educational Statistics: Statistical Terms and Vocabulary Internal vs External Validity Internal validity: concerned with whether the methods and procedures used in a study warrant the conclusions drawn from the study. External validity: concerned with the extent to which the results obtained on a sample generalize to the target population.

Educational Statistics: Statistical Terms and Vocabulary Statistical procedures are the tools of research. There are several types (or methods) of research studies and the type of statistical procedure used will often vary from one type of research to another.

Educational Statistics: Statistical Terms and Vocabulary The correlational method of research. • Examines relationships among two or more variables. • For example: What is the relationship between hours of TV watched per day and the number of calories consumed per day? – Note that there no cause-effect relationship is postulated. – Correlation does not imply causation.

Educational Statistics: Statistical Terms and Vocabulary The experimental method is used when the researchers wants to establish a cause and effect relationship. • The researcher manipulates one variable (the independent) variable, and • Observes (or measures) what happens to a second variable (the dependent variable), • While Controlling for all other variables (extraneous variables).

Educational Statistics: Statistical Terms and Vocabulary A quasi-experiment is similar to a (true) experiment except that here the independent variable is not manipulated by the researcher. For example, in studying the effects of sex on mathematics achievement a researcher compares boys and girls (the independent variable).

Educational Statistics: Measurement • Another tool of [quantitative] research. • Definition: A rule for the assignment of numbers to attributes or characteristics of individuals, or things. • Eg. : – 1 if Male, 2 if Female. – Score on a test. – A judge’s rating (on a scale of 1 to 10) of physical attractiveness.

Educational Statistics: Scales of Measurement Types of measurement. • The type of measurement scale has implications for the type of statistical procedure employed. • Some statistical procedures assume a certain level of measurement. • Three types of measurement can be distinguished: nominal, ordinal, and scale (includes interval and ratio).

Educational Statistics: Scales of Measurement Types of measurement: nominal. • Coarse level of measurement used for identification purposes. • Substitutes numbers for other categorical labels. • No order of magnitude is implied. • Examples: sex (male or female), student classification (freshman, sophomore, junior, senior), etc. • Sometimes referred to a qualitative measurement.

Educational Statistics: Nominal-scale Data • Also called categorical or qualitative data (or variables). • Represent lowest level of measurement. • Classify individuals into one of two or more mutually exclusive categories. • The categories are usually represented by numbers. Eg: – 1 if Male, 2 if Female. – 1 if Democrat, 2 if Republican, 3 if Independent, 4 if Other. • The numbers DO NOT indicate more or less of an attribute.

Educational Statistics: Scales of Measurement Types of measurement: ordinal. • Objects measured on an ordinal scale differ from each other in terms of magnitude, but the units of magnitude are not equal. • The objects can be ordered in terms of their magnitude (more or less of an attribute. • Examples: class rank, seeding in golf or tennis, percentiles, level of motivation. • Do not allow common mathematical operations.

Educational Statistics: Ordinal-scale Data • In addition to classifying individuals, ordinal scales rank individuals in terms of the degree to which they possess measures characteristics of attributes. • Ordinal scales allow us to compare individuals in terms of who has more (or) less of a characteristic or attribute. • Ordinal scales do NOT indicate HOW MUCH more or less. • Eg. : – Class rank – Acrobatic competition judgments. • What about grades or GPA?

Educational Statistics: Scaled Data • Includes both interval and ratio level scales. • Scale measurements yield equal intervals between adjacent scale points. • The difference between 5’-6” and 6’ is the same as the difference between 3’ and 3’-6”. • The difference between an SAT-V score of 435 and 445 is the same as the difference between a score of 520 and 530. • Most scores obtained form achievement tests, aptitude tests, etc. are treated as scaled data.

Educational Statistics: Importance of the Type of Measurement • We will see, shortly, that the type of scale assumed in measuring manifestations of variables has implications for the type of statistical procedure used. • At the end of the day, however, the type of scale assumed is less important than the interpretations we give to the manipulations of the numbers we obtain when we measure a variable.

Educational Statistics: Variables • Any event, category, behavior, or attribute that can: – take on different values, and – can be measured. • Examples: age test score class size type of instruction achievement group assignment motivation size of print creativity

Educational Statistics: Types of Variables Discrete and Continuous variables • Variables can also be described in terms of the types of values they can be assigned. – Discrete variables are categorical. No values between two adjacent values are permissible. – Continuous variables can (theoretically) have an infinite number of values.

Educational Statistics: Types of Variables • • • Independent variables. Dependent variables. Extraneous variables. Confounding variables. Intervening variables.

Educational Statistics: Independent Variables • True independent variables: – Experimental. – Manipulated. – Controlled. • Quasi-independent variables: – Naturally occurring. – Organic or biological. – Quasi-experimental

Educational Statistics: Dependent Variables • • Effects. Outcomes. Measured variables. Dependent variables are functions of independent variables.

Educational Statistics: Extraneous Variables • • Nuisance or controlled variables. Irrelevant to the focus of the study. Can affect interpretation of results. Examples: time of day sequence of events side of building sex of investigator age of school building current events

Educational Statistics: Confounding Variables • Extraneous variables whose effects on the dependent variables cannot be distinguished from those of the independent variable(s). • Usually occurs when an extraneous variable is correlated with one or more independent variables.

Educational Statistics: Intervening Variables • “Black box” variables. • Invented to account for internal, unobservable psychological processes that intervene between independent and dependent variables. • E. g. learning intervenes between teaching and achievement:

Educational Statistics: Continuous Variables • Numerical data in research can be classified as either continuous or discrete. • Variables that can take on any of a continuously ordered set of values within some specified range. • Examples age experience achievement dogmatism motivation intelligence

Educational Statistics: Discrete Variables • Variables whose values can only be whole numbers. • Characterized by gaps in the measurement scale. • Typically represent counts of things. number of children size of family school enrollment number of books

Educational Statistics: Continuous or Discrete: Can you tell? • How would you classify the following variables? Continuous or discrete? Grade Level Occupation College classification Time on task • Actually it depends upon how these variables are defined. – What is the underlying characteristic or trait? – Is it continuous or naturally discrete?

Educational Statistics: Quantitative vs Qualitative Data • Quantitative data (sometimes called measurement data) results from some form of measurement. – Typically involves the use of some type of measuring instrument. • Qualitative data (also called categorical data or frequency data) – results from counting the occurrences of variables.

Educational Statistics: The END!